Base Numbers

Because they are on the same bases, we can disregard them. 

Now, we have the equation: $AAA + BBB + 111 = 666$

This means $A + B = 5$

The only possible cases are $(1,4)$, $(2,3)$, $(3,2)$, and $(4,1)$. Thus, are only $\color{brown}\boxed{4}$ cases that